Mathbox for Stefan O'Rear |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > isnumbasgrplem3 | Structured version Visualization version GIF version |
Description: Every nonempty numerable set can be given the structure of an Abelian group, either a finite cyclic group or a vector space over Z/2Z. (Contributed by Stefan O'Rear, 10-Jul-2015.) |
Ref | Expression |
---|---|
isnumbasgrplem3 | ⊢ ((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) → 𝑆 ∈ (Base “ Abel)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hashcl 13009 | . . . . . 6 ⊢ (𝑆 ∈ Fin → (#‘𝑆) ∈ ℕ0) | |
2 | 1 | adantl 481 | . . . . 5 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → (#‘𝑆) ∈ ℕ0) |
3 | eqid 2610 | . . . . . 6 ⊢ (ℤ/nℤ‘(#‘𝑆)) = (ℤ/nℤ‘(#‘𝑆)) | |
4 | 3 | zncrng 19712 | . . . . 5 ⊢ ((#‘𝑆) ∈ ℕ0 → (ℤ/nℤ‘(#‘𝑆)) ∈ CRing) |
5 | crngring 18381 | . . . . 5 ⊢ ((ℤ/nℤ‘(#‘𝑆)) ∈ CRing → (ℤ/nℤ‘(#‘𝑆)) ∈ Ring) | |
6 | ringabl 18403 | . . . . 5 ⊢ ((ℤ/nℤ‘(#‘𝑆)) ∈ Ring → (ℤ/nℤ‘(#‘𝑆)) ∈ Abel) | |
7 | 2, 4, 5, 6 | 4syl 19 | . . . 4 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → (ℤ/nℤ‘(#‘𝑆)) ∈ Abel) |
8 | hashnncl 13018 | . . . . . . . 8 ⊢ (𝑆 ∈ Fin → ((#‘𝑆) ∈ ℕ ↔ 𝑆 ≠ ∅)) | |
9 | 8 | biimparc 503 | . . . . . . 7 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → (#‘𝑆) ∈ ℕ) |
10 | eqid 2610 | . . . . . . . 8 ⊢ (Base‘(ℤ/nℤ‘(#‘𝑆))) = (Base‘(ℤ/nℤ‘(#‘𝑆))) | |
11 | 3, 10 | znhash 19726 | . . . . . . 7 ⊢ ((#‘𝑆) ∈ ℕ → (#‘(Base‘(ℤ/nℤ‘(#‘𝑆)))) = (#‘𝑆)) |
12 | 9, 11 | syl 17 | . . . . . 6 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → (#‘(Base‘(ℤ/nℤ‘(#‘𝑆)))) = (#‘𝑆)) |
13 | 12 | eqcomd 2616 | . . . . 5 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → (#‘𝑆) = (#‘(Base‘(ℤ/nℤ‘(#‘𝑆))))) |
14 | simpr 476 | . . . . . 6 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → 𝑆 ∈ Fin) | |
15 | 3, 10 | znfi 19727 | . . . . . . 7 ⊢ ((#‘𝑆) ∈ ℕ → (Base‘(ℤ/nℤ‘(#‘𝑆))) ∈ Fin) |
16 | 9, 15 | syl 17 | . . . . . 6 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → (Base‘(ℤ/nℤ‘(#‘𝑆))) ∈ Fin) |
17 | hashen 12997 | . . . . . 6 ⊢ ((𝑆 ∈ Fin ∧ (Base‘(ℤ/nℤ‘(#‘𝑆))) ∈ Fin) → ((#‘𝑆) = (#‘(Base‘(ℤ/nℤ‘(#‘𝑆)))) ↔ 𝑆 ≈ (Base‘(ℤ/nℤ‘(#‘𝑆))))) | |
18 | 14, 16, 17 | syl2anc 691 | . . . . 5 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → ((#‘𝑆) = (#‘(Base‘(ℤ/nℤ‘(#‘𝑆)))) ↔ 𝑆 ≈ (Base‘(ℤ/nℤ‘(#‘𝑆))))) |
19 | 13, 18 | mpbid 221 | . . . 4 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → 𝑆 ≈ (Base‘(ℤ/nℤ‘(#‘𝑆)))) |
20 | 10 | isnumbasgrplem1 36690 | . . . 4 ⊢ (((ℤ/nℤ‘(#‘𝑆)) ∈ Abel ∧ 𝑆 ≈ (Base‘(ℤ/nℤ‘(#‘𝑆)))) → 𝑆 ∈ (Base “ Abel)) |
21 | 7, 19, 20 | syl2anc 691 | . . 3 ⊢ ((𝑆 ≠ ∅ ∧ 𝑆 ∈ Fin) → 𝑆 ∈ (Base “ Abel)) |
22 | 21 | adantll 746 | . 2 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ 𝑆 ∈ Fin) → 𝑆 ∈ (Base “ Abel)) |
23 | 2nn0 11186 | . . . . . . 7 ⊢ 2 ∈ ℕ0 | |
24 | eqid 2610 | . . . . . . . 8 ⊢ (ℤ/nℤ‘2) = (ℤ/nℤ‘2) | |
25 | 24 | zncrng 19712 | . . . . . . 7 ⊢ (2 ∈ ℕ0 → (ℤ/nℤ‘2) ∈ CRing) |
26 | crngring 18381 | . . . . . . 7 ⊢ ((ℤ/nℤ‘2) ∈ CRing → (ℤ/nℤ‘2) ∈ Ring) | |
27 | 23, 25, 26 | mp2b 10 | . . . . . 6 ⊢ (ℤ/nℤ‘2) ∈ Ring |
28 | eqid 2610 | . . . . . . 7 ⊢ ((ℤ/nℤ‘2) freeLMod 𝑆) = ((ℤ/nℤ‘2) freeLMod 𝑆) | |
29 | 28 | frlmlmod 19912 | . . . . . 6 ⊢ (((ℤ/nℤ‘2) ∈ Ring ∧ 𝑆 ∈ dom card) → ((ℤ/nℤ‘2) freeLMod 𝑆) ∈ LMod) |
30 | 27, 29 | mpan 702 | . . . . 5 ⊢ (𝑆 ∈ dom card → ((ℤ/nℤ‘2) freeLMod 𝑆) ∈ LMod) |
31 | lmodabl 18733 | . . . . 5 ⊢ (((ℤ/nℤ‘2) freeLMod 𝑆) ∈ LMod → ((ℤ/nℤ‘2) freeLMod 𝑆) ∈ Abel) | |
32 | 30, 31 | syl 17 | . . . 4 ⊢ (𝑆 ∈ dom card → ((ℤ/nℤ‘2) freeLMod 𝑆) ∈ Abel) |
33 | 32 | ad2antrr 758 | . . 3 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → ((ℤ/nℤ‘2) freeLMod 𝑆) ∈ Abel) |
34 | eqid 2610 | . . . . . . 7 ⊢ (Base‘((ℤ/nℤ‘2) freeLMod 𝑆)) = (Base‘((ℤ/nℤ‘2) freeLMod 𝑆)) | |
35 | 24, 28, 34 | frlmpwfi 36686 | . . . . . 6 ⊢ (𝑆 ∈ dom card → (Base‘((ℤ/nℤ‘2) freeLMod 𝑆)) ≈ (𝒫 𝑆 ∩ Fin)) |
36 | 35 | ad2antrr 758 | . . . . 5 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → (Base‘((ℤ/nℤ‘2) freeLMod 𝑆)) ≈ (𝒫 𝑆 ∩ Fin)) |
37 | simpll 786 | . . . . . 6 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → 𝑆 ∈ dom card) | |
38 | numinfctb 36692 | . . . . . . 7 ⊢ ((𝑆 ∈ dom card ∧ ¬ 𝑆 ∈ Fin) → ω ≼ 𝑆) | |
39 | 38 | adantlr 747 | . . . . . 6 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → ω ≼ 𝑆) |
40 | infpwfien 8768 | . . . . . 6 ⊢ ((𝑆 ∈ dom card ∧ ω ≼ 𝑆) → (𝒫 𝑆 ∩ Fin) ≈ 𝑆) | |
41 | 37, 39, 40 | syl2anc 691 | . . . . 5 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → (𝒫 𝑆 ∩ Fin) ≈ 𝑆) |
42 | entr 7894 | . . . . 5 ⊢ (((Base‘((ℤ/nℤ‘2) freeLMod 𝑆)) ≈ (𝒫 𝑆 ∩ Fin) ∧ (𝒫 𝑆 ∩ Fin) ≈ 𝑆) → (Base‘((ℤ/nℤ‘2) freeLMod 𝑆)) ≈ 𝑆) | |
43 | 36, 41, 42 | syl2anc 691 | . . . 4 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → (Base‘((ℤ/nℤ‘2) freeLMod 𝑆)) ≈ 𝑆) |
44 | 43 | ensymd 7893 | . . 3 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → 𝑆 ≈ (Base‘((ℤ/nℤ‘2) freeLMod 𝑆))) |
45 | 34 | isnumbasgrplem1 36690 | . . 3 ⊢ ((((ℤ/nℤ‘2) freeLMod 𝑆) ∈ Abel ∧ 𝑆 ≈ (Base‘((ℤ/nℤ‘2) freeLMod 𝑆))) → 𝑆 ∈ (Base “ Abel)) |
46 | 33, 44, 45 | syl2anc 691 | . 2 ⊢ (((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆 ∈ Fin) → 𝑆 ∈ (Base “ Abel)) |
47 | 22, 46 | pm2.61dan 828 | 1 ⊢ ((𝑆 ∈ dom card ∧ 𝑆 ≠ ∅) → 𝑆 ∈ (Base “ Abel)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∩ cin 3539 ∅c0 3874 𝒫 cpw 4108 class class class wbr 4583 dom cdm 5038 “ cima 5041 ‘cfv 5804 (class class class)co 6549 ωcom 6957 ≈ cen 7838 ≼ cdom 7839 Fincfn 7841 cardccrd 8644 ℕcn 10897 2c2 10947 ℕ0cn0 11169 #chash 12979 Basecbs 15695 Abelcabl 18017 Ringcrg 18370 CRingccrg 18371 LModclmod 18686 ℤ/nℤczn 19670 freeLMod cfrlm 19909 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-seqom 7430 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-ec 7631 df-qs 7635 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-acn 8651 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-hash 12980 df-dvds 14822 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-0g 15925 df-prds 15931 df-pws 15933 df-imas 15991 df-qus 15992 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-nsg 17415 df-eqg 17416 df-ghm 17481 df-gim 17524 df-gic 17525 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-oppr 18446 df-dvdsr 18464 df-rnghom 18538 df-subrg 18601 df-lmod 18688 df-lss 18754 df-lsp 18793 df-sra 18993 df-rgmod 18994 df-lidl 18995 df-rsp 18996 df-2idl 19053 df-cnfld 19568 df-zring 19638 df-zrh 19671 df-zn 19674 df-dsmm 19895 df-frlm 19910 |
This theorem is referenced by: isnumbasabl 36695 dfacbasgrp 36697 |
Copyright terms: Public domain | W3C validator |